8. If n̂ is the unit normal vector (with non-negative z...

8. If \hat{n} is the unit normal vector (with non-negative $z$ component) to the surface $S$ at $\vec{r}(\frac{\pi}{4},0)$, then which of the following vectors is in the same direction as \hat{n}? a. $(\sqrt{2},0,\sqrt{2})$ b. $(-\sqrt{2},0,\sqrt{2})$ c. $(-1,0,3)$ d. $(1,0,3)$ e. $(1,0,\sqrt{2})$

Transcript

00:01 So we have f x y z is equal to zero and we want to find the unit normal vector which i'll call n hat at the point two negative one negative nine or i'm sorry they actually call it n arrow.

00:26 Okay so the unit normal vector is equal to the gradient of f divided by its magnitude.

00:34 Okay and the gradient of f is just equal to the partial derivative, i'm sorry we need to evaluate this at the point also but we'll do that later.

00:46 So we have the derivative partial derivative of f with respect to x with respect to y and with respect to z.

00:53 Okay and then we just divide by the square root of the partial derivative squared each partial derivative squared sum together.

01:03 Okay because that's the magnitude of this vector.

01:09 So let's start by finding the partial derivative of f with respect to x.

01:13 We get one divided by what's inside the parentheses here so we can just take the reciprocal so we have five y minus z divided by two x and then we need to multiply by the derivative of what's inside the parentheses with respect to x which is just two divided by five y minus z.

01:38 So we see that five y minus z cancels and the twos cancel and we are left with one over x.

01:45 When we evaluate this at the point so i'll say f x at p okay we plug in x equals two so we have one half.

01:57 Okay now let's find the partial derivative with respect to y.

02:00 So again we need to take the reciprocal of what's inside the parentheses so we have five y minus z divided by two x and now we'll take the derivative of what's inside the parentheses so we get negative two x over five y minus z squared.

02:24 Okay multiplied by the derivative of five y minus z which is five.

02:31 Okay so we can divide out one factor of five y minus z and we can divide out the two x.

02:40 Okay so then we're left with negative five divided by five y minus z.

02:47 When we evaluate this at the point we need to plug in y equals negative one and z is negative nine so we get negative five divided by five times negative one minus a negative nine...

Question

8. If \hat{n} is the unit normal vector (with non-negative $z$ component) to the surface $S$ at $\vec{r}(\frac{\pi}{4},0)$, then which of the following vectors is in the same direction as \hat{n}? a. $(\sqrt{2},0,\sqrt{2})$ b. $(-\sqrt{2},0,\sqrt{2})$ c. $(-1,0,3)$ d. $(1,0,3)$ e. $(1,0,\sqrt{2})$

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